Article ID: MTJPAM-D-20-00001

Title: Non-Linear Effects in Incompressible Viscous Unidirectional Fluid Flows

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00001; Volume 2 / Issue 1 / Year 2020, Pages 1-35

Document Type: Research Paper

Author(s): Alexey Zhirkin a

aNational Research Center “Kurchatov Institute” (NRC KI), Kurchatov Complex of Fusion Power and Plasma Technologies, Fusion Reactor Department, Reactor Problem Laboratory, Academician Kurchatov square 1, Moscow 123182, Russia

Received: 28 January 2020, Accepted: 12 March 2020, Available online: 9 May 2020.

Corresponding Author: Alexey Zhirkin (Email address:

Full Text: PDF


More accurate nonlinear equations for the divergence free velocity field are obtained by considering small dissipation due to inelastic collisions in the three-dimensional Navier-Stokes equations. The approach of fluid incompressibility is not broken. The modified equations are used within the boundary layer. The nonlinear solutions obtained for the Couette and Poiseuille flow explain the paradoxes of symmetry and turbulence of viscous fluid flow. Laminar-turbulent transition of a steady and unsteady flow is analyzed. The process of space and time (blow-up) symmetry breaking of the Cauchy problem solution for the homogeneous and non-homogeneous Navier-Stokes equations is described. The dynamics of solitary waves in a dissipative medium without dispersion is investigated. Dissipative forms of the Korteweg-de Vries and Korteweg-de Vries-Burgers equation in one and two spatial dimensions as well as the Kadomtsev-Petviashvili equation is derived. Solitary wave solutions of these equations are presented as dissipative structures rather than wave packets. Solutions of the modified equations reveal the dynamo effect for a plane-parallel incompressible viscous hydro magnetic fluid flow. The results of the research show that the used approach is quite productive.

Keywords: Partial differential equations of mathematical physics, Navier-Stokes equations, Boltzmann equation in fluid mechanics, incompressible viscous fluid, boundary-layer theory, separation, transition to turbulence, waves for incompressible viscous fluids, instability of magneto-hydrodynamic flows

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